Pre-classical, adhoc-but-easy method of converting an analog prototype filter to a digital IIR filter. Does not preserve any optimality.

Impulse invariance means that digital filter impulse
response exactly equals samples of the analog prototype impulse
response:

The impulse response of a causal, stable analog filter
is simply a sum of decaying exponentials:

This technique is used occasionally in digital simulations of analog filters.

What is the main problem/drawback with this design technique?

Since it samples the non-bandlimited impulse response of the analog prototype filter, the frequency response aliases. This distorts the original analog frequency and destroys any optimal frequency properties in the resulting digital filter.